I need a better understanding of notation for quantifiers

Question

The statement:

(∀x ∈ Z) ((∃y ∈ Z) x = 2y) or ((∃y ∈ Z)x = 2y+1)

says that every integer is even or odd.

I can break down the statement into each part (∀x ∈ Z) means for all x in set Z, (∃y ∈ Z) x = 2y) is at least one integer y in set Z, x = 2y, so on and so on, but I don't understand how it is read as a whole.

I'm asking if anyone can break down the reading process of each statement leading up to the overall answer for a better understanding to how it is meant to be read to get the final answer "every integer is even or odd".

Answer 1

In plain english: Any integer number is twice another integer number or its successive number is twice an integer number (, or both).

Where I addes the or both to emphasize that the mathematical or is non-exclusive (that would be xor)

Or equivalently, by definition of even and odd:

Any integer number is even or odd (, or both).

Answer 2

I find it helps to read "$\exists$" as "we can find," and $\forall$ as "for each." While not technically true, as there may be no algorithm for finding the thing, it helps straighten out the difference between $\exists, \forall$ and $\forall, \exists$.

The first says "We can find a thing, which for all other things, blah blah blah blah."

The second says "for each such and such, we can find a this or that, ......"

In your example, this says, "for every integer, we can find another integer, so that the first can be expressed in one of these two forms."